Question

Let the test statistic T have a t distribution when H0 is true. Give the significance level for each of the following situation.

A. Ha: mu > m0, df = 15, rejection region t > 3.733
B. Ha : mu < mu 0, n = 24, rejection region t < - 2.500
C. Ha: mu not = mu 0, n = 31, rejection region t >1.697 or t < - 1.697

Answer 1

a) The degrees of freedom are given by:

`df = 15`

And the rejection region is
`t_(\alpha)<3.733`

And the significance level would be:

`P(t_(15) >3.733) =0.001`

b) The degrees of freedom are given by:

`df = 24-1=23`

And the rejection region is
`t_(\alpha) <-2.5`

And the significance level would be:

`P(t_(23) <-2.5) =0.0099 \approx 0.01`

c) The degrees of freedom are given by:

`df = 31-1=30`

And the rejection region is
`t_(\alpha) <-1.697` or
`t_(\alpha) >1.697`

And the significance level would be:

`2*P(t_(30) <-1.697) =0.10`

Explanation:

Part a

We have the following system of hypothesis:

Null hypothesis:
`\mu \leq \mu_0`

Alternative hypothesis:
`\mu > \mu_0`

The degrees of freedom are given by:

`df = 15`

And the rejection region is
`t_(\alpha)<3.733`

And the significance level would be:

`P(t_(15) >3.733) =0.001`

Part b

We have the following system of hypothesis:

Null hypothesis:
`\mu \geq \mu_0`

Alternative hypothesis:
`\mu < \mu_0`

The degrees of freedom are given by:

`df = 24-1=23`

And the rejection region is
`t_(\alpha) <-2.5`

And the significance level would be:

`P(t_(23) <-2.5) =0.0099 \approx 0.01`

Part c

We have the following system of hypothesis:

Null hypothesis:
`\mu = \mu_0`

Alternative hypothesis:
`\mu \\eq \mu_o`

The degrees of freedom are given by:

`df = 31-1=30`

And the rejection region is
`t_(\alpha) <-1.697` or
`t_(\alpha) >1.697`

And the significance level would be:

`2*P(t_(30) <-1.697) =0.10`